# Nonlinear Stochastic Optimal Control and Optimal Stopping using the Fokker-Planck Transformation

arXiv: https://arxiv.org/abs/2604.12153 · HTML: https://mlsanity.com/paper/2604.12153/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2026.

## Built on by (most cited first, 1 of 1)

- [Lifted Schr"odinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions](https://mlsanity.com/paper/2605.24795/) (2026)

## Abstract (arXiv)

In this paper, we develop a theoretical framework for nonlinear stochastic optimal control problems with optimal stopping by establishing a density-based deterministic representation of the underlying diffusion. For state-independent diffusion, we rewrite the controlled Fokker-Planck equation as a continuity equation driven by a score-corrected velocity field, yielding a deterministic characteristic dynamics that reproduces the marginal law of the stochastic system. Leveraging Stein-type identities, we show that the associated distributional dynamic programming equation admits the same second-order differential operator as the distributional stochastic Hamilton-Jacobi-Bellman formulation. Building on this representation, we formulate an optimal control problem with state-dependent terminal-time assignment and terminal distributional constraints and derive the first-order necessary conditions using variational analysis. We present the conditions both for a common terminal time and for the general case of state-dependent stopping.